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Monday, August 23, 2010

Testing the foundations of quantum mechanics

If you know one thing about quantum mechanics, it's Born's rule: The probability of a measurement is the square of the amplitudes of the wave-functions. It is the central axiom of quantum mechanics and what makes it quantum. If you have a superposition of states, the amplitudes are sums of these states. Taking the square to obtain the probability means you will not only get the square of each single amplitude - which would be the classical result - but you will get mixed terms. These mixed terms are what is responsible for the interference in the famous double-slit experiment and yield the well-known spectrum with multiple maxima rather than one reproducing the two slits, as you'd get were the particles classical. (Dr. Quantum shows you what I mean.)

The short summary is that they haven't found any deviation to a precision of one in a hundred. But their method is really neat and worth spending a paragraph on.

The experimental setup that the group has used is a tripe-slit through which pass single photons. If one computes the probability to measure a photon at a particular location on the detector screen in usual quantum mechanics, you square the sum of the wave-functions originating from each of the three slits. You get several mixed terms, but they are all second order in the wave-function. If Born's rule holds, this allows you to express the probability for the three-slit experiment as a sum of probabilities from leaving open only one of the slits and leaving open combinations of two slits. Thus, what the clever experimentalist do is a series of measurements leaving each single slit open, all combinations of two slits open, and leaving all three slits open, and see if the probabilities add up. And they do, to very good precision.

So, there's nothing groundbreaking to report here in terms of novel discoveries, but I very much like the direct test of the foundations of quantum mechanics this experiment constitutes. I think we could use more tests in this direction, and higher precision will come with time.

51 comments:

This enterprise is very delicate, since if they had obtained a "positive" result, they would then have to do no less than "prove" that no possible prepared state and POV measurement could have resulted in whatever they observed. In the experimentalist language of their article, they say "With a null experiment, a very careful analysis of random and systematic errors must be undertaken". The tradeoffs in adopting a higher order modification of Born's rule so that we can keep a simpler system of states and measurements to describe experiments and their results are more delicate than this Science article has space to discuss.

From a Duhem-like perspective, the statement quoted above is an admission that a test is not a test of a single feature of a theory, but a test of an overall theory or system of models. In this case, the test is of how careful their "analysis of random and systematic errors" has been, as well as of whether Born's rule is correct. As always, your post hints at this kind of subtlety, but I think it is worth spelling out this aspect of the original paper.

Attempting this kind of experiment teases out details that are of great interest, but I think your final point that increasing precision will come would be well-tempered by an equal emphasis on completeness of analysis.

So as a layman, the question for me is also about speaking to the foundation of quantum mechanics as it forms the basis of entanglement and it's progression?

Max Born's statistical interpretation of the wave function ended determinism in atomic world. These men - Bohr, Heisenberg, Kramers, Dirac and Born together with Born represent the founding fathers of quantum mechanics. Louis de Broglie wrote his dissertation on the wave nature of matter which Schrodinger used as basis for wave mechanics. Albert Einstein whose famous response to Born's statistical interpretation of wave function was "God does not play dice."http://www.freesciencelectures.com/ --- Notice: This video is copyright by its respectful owners. The website address on the video does not mean anything.

I bet few are surprised at these results, as ... yet again, QM has been experimentaly verified. I was very disappointed at how easy the mathematics of Quantum Mechanics came to me. Then again, before I sudied QM (a requirement to obtain a BSME, and a BSEE I'm sure ... not sure about a BSChE or a Bachelor's in Civil Engineering ... really, how much microstuff do you have to know to design a dam or a bridge?), I had 4 semesters of Calculus and a strong dose of the Classical Physics stuff, and a 5th potpouri math course in Linear Algebra, Fourier series, Laplace tranforms (very nice), and some cool curves called Gamma and Beta functions.

In other words ... I like a challenge! QM fails to provide that. Interpretations of QM tend more to the Philosophical than the Physical, with obvious repercussions for the Physical. But I don't worry about that, as future technology will allow deeper probing into smaller and smaller intervals of time and length ... and/or the freaky "extra" (extraneous?) dimensions superstringers and the science jouranlists that adore them atempt to tell us exist.

Quantum Field Theory is a nice challenge though. Much work to be done there. Gen Rev and Black Hole Thermodynamics too. Quantum Gravity, yum, except Martin Gleisner has me questioning if there even IS a QG theory ... since he questions the existance of a T.O.E., why stop there?

Just wondering how Feynman would have analyzed those photos of Einstein Rings. It sure looks like the photons were going along geodesic paths, while the universe was expanding, and that Einstein's geometrical approach would be the simplest explanation, but one would have to entertain paths that can be bent and stretched. viz Plato's remark. Just curious

Yes, you're right of course. It's also not a priori clear, at least not tome, that there couldn't be a modification of Born's rule that still allows this particular sum rule for the triple-slit. Though the most plausible modification, an additional higher order term (with some small pre-factor) wouldn't do I think. (You'd never get an ABC term from the double-slit contribution, but you'd have such a term in the triple-slit probability.)

Actually, a precision of one in a hundred is not exactly a very high precision. Still, better than nothing at all. But of course it remains the question, exactly as you say, had they found something, what really would we conclude from it? I suppose most likely what we'd conclude is that more experiments were needed, most notably different setting to single down the origin of the deviations. Best,

Not sure what you're asking for. Do I have knowledge in regards to entanglement. Well, I know what it is. Has it been tested? It's been confirmed and used in a long list of experiments without there being any deviation from standard quantum mechanics. "Entanglement" however, in contrast to Born's rule, is not an axiom of quantum mechanics, rather, it's a consequence. Best,

Have you made your way though representation theory of Lie-groups, and in particular angular momentum coupling? I hated that stuff. Whenever somebody says Clebsch-Gordan, I run for the door ;-) You can spend, it seems to me, an infinite amount of time with Young tableaus. In any case, the representation theory stuff is very rich and goes very deep through both quantum mechanics and quantum field theory, so if you're looking for a challenge... Best,

As with Green's function and Newtonian gravitation (not shown in GR), Born's squaring removes configuration chirality. Gravitation and quantum mechanics so perfectly portraying white swans are fundamentally blind to black swans. A filled sphere in 3-space is exhaustively, definitively not geometrically chiral. This can be compromised.

Yes, I meant Marcelo Gleiser. I subconciously "Americanized"/"Anglocized" his name, like the clerk at Ellis Island circa 1902 who magically changed (with the stroke of a pen) my grandfather Stefan's birthname to Steven. My bad, and thanks for the correction.

Hi Bee,

No I haven't studied that stuff. Dammit Bee, I'm an Engineer, not a Physicist! lol

We're the guys who work on the stuff you guys pass on to us when the math and experimental results are a done deal ... stuff like Thermodynamics and Quantum Mechanics.

And then the Policitians have to play catchup, which is important because THEY dispense the money.

Think of President Obama being amazed and impressed with power plant Steam Turbine-Generator sets ... which is REALLY state-of-the-art stuff ... if the year is 1910. :-)

What I really enjoyed about the paper is how simple it was to read. This is not true of many papers I read.

I've recently been thinking about how little I know about the experimental verifications our theories. I can rattle off the classical experiments for general relativity and some for quantum mechanics, but I really don't understand them at all.

One book I really enjoyed was Purcell's E&M book since he constantly put in references and descriptions of papers that did tests on various parts of E&M theory. I believe he starts off the book with the problem of proton and electron charge. Are they really the same? He then describes an experiment that verifies it to within some very small order of magnitude.

Actually, I found it a bit confusing that the paper explicitly states the Born rule in the first equation, as this made me wonder what additional "third-order" term one possibly could expect...

On the other hand, writing down a formal expansion of the detection probability (not the amplitude) as a sum of one-slit probabilities, two-slit probabilities, three-slit probabilities, and so on, is an intriguing idea - the paper by Rafael Sorkin credited for this is on the arxiv (gr-qc/9401003).

In classical mechanics, adding one-slit contributions does the job, while in "standard" quantum mechanics, one-slit and two-slit contributions are "complete". But in principle, it might be possible that higher contributions are relevant - and just these contributions are constrained by this experiment.

I am wondering, is there a theorem that if the expansion "stops" at the two-slit contribution, then this necessarily implies the Born rule?

I have in mind here a vague analogy to the moments of a probability distribution, where, when all even moments beyond the second can be expressed by the second moment, this implies a Gaussian distribution, which has the square modulus in the exponent...

The Born Rule is about "statistics", basically in terms of yes/no results in a detector (like hit or not, passed x-polarization or not.) All that is closely related to the projection postulate. (Note the provisonality implied by that phrase.) The idea of weak measurements and similar implies that we can find out more than just those binary answers. If so it would not actually violate the BR. It would be like, instead of 64% chance of a hit given a net amplitude of 0.8, we could actually estimate the amplitude itself as say, 90% chance of being between 0.75 and 0.85.

Some people like Y. Aharonov (of that famous Aharonov-Bohm effect) think we can find out more (not quite in the manner I described anyway, AFAICT), others (more traditional) think not. Proponents sometimes use the term "weak measurements." I admit to not being quite sure of their arguments or how radical it would be. One of those parameters is photon circularity. I presented my own proposal to do that in a post "Proposal summary:..." at name link. (Hopefully on-top' enough to be worth noting.)

I don't think anyone has yet accomplished either a firm theoretical proof it is possible, or experimental example. It seems to me that maybe QM needs some ambiguity in the actual wave function (ie, to hide all but the actual binary chances) or else it could cause mischief about the polarization of two entangled photons. Actual characterization of both polarizations (rather than the simple demand for correlation of results), would lead to an issue like this: Say I find that both are "at 12 degrees" polarization. Well, then how do I explain correlated hits using polarizers at zero angle? The chance should have been cos^2(12 degrees) for each. The whole entanglement description depends on the photons not having a specific, independent individual ("actual polarization") angle, even of the WF.

Yet it seems odd to be able to create one photon at a specific polarization angle (easy, just through that corresponding polarizer) which and it will definitely make a hit in the corresponding detector. Furthermore that photon has a definite individual representation in term of basis composition (unlike, we suppose, an entangled photon?) But supposedly we can't find this out by studying that photon after creation. So there is a property, that an insider can use to prove his knowledge, but a naive researcher has to just try an angle and see if he gets a hit (according to BR statistic.) That seems contradictory. Yet if we can find out more, that would certainly challenge the foundations of QM.

It is sad that the nice artwork that went into the Dr. Quantum video comes from the unmentionable perversion of science known as "What the bleep do we know". Though the video is not too bad when seen in isolation.

I'll add to my earlier post. Take Born's rule to be that V_{ij}=Tr(rho_j O_i) --- that is, the value V_{ij} of a specific statistic of a concrete experimental dataset that is associated with the state with the j-th density matrix rho_j and the i-th observable O_i is Tr(rho_j O_i). We only know what state and observable we are measuring by the results we obtain --- that is, we solve the above (nonlinear) equation for the matrices rho_j and O_i, given the (finite set of) experimental statistics. If we obtain experimental statistics that have no solution for a given dimensionality of Hilbert space, we can always introduce a Hilbert space for which there is a solution. [Note that "solution" in the sense of a very good approximation is almost always acceptable, in which case a larger Hilbert space will always allow for better accuracy.]

At this level of theory, which is as close to "just the Born rule" as I think we can get, QM can accommodate any experimental data whatsoever. It is not falsifiable. If we limit the dimensionality of Hilbert space we allow ourselves to use, however (or what density matrices or measurement operators we allow in a given experimental context), we might constrain the problem to have no solution, but that only invalidates the Born rule relative to that larger theory and modeling environment.

In practice, many rules of art are applied in the choice of what ansätze to use when modeling a given class of experiments, some of which can be clearly seen in the article in Science that is discussed here. The choice between elements of that relatively uncodified set of rules and the single Born rule is rather asymmetric. I think much more discussion is needed even to begin to justify modifying the Born rule.

As Stefan, I found the paper to be somewhat confusing in that they don't explain what modification it is they're actually testing. They just write down the normal Born rule, and then the probability with a possible modification, but it left me wondering well, what's the theory that leads to such a modification. One should probably look at Sorkin's paper for that.

Far worse than the paper was actually the "Perspective" on the article in the same Science issue by Fanson, Pairs Rule Quantum Interference. He keeps talking in his article about "interactions" between paths that lead to additional interference terms. I found it confusing because there's nothing "interacting" there, and certainly not paths, and in any case the photons always take all paths anyway, so that explanation is absolutely not helpful. Best,

Thanks for your synopsis in respect to this recent testing of the foundational principles of QM in more complex situations and certainly represents the type of experiments I’d like to see more of. I’m particularly intrigued by the group’s suggestion found at the end of their paper to extend such experiments to particles that have an actual rest mass such as neutrons to see if deviations from the Born rule can be identified. That is I would be curious if the actions of particles for whom time and distance is meaningful would make any difference. Also I’m grateful to Stefan for pointing out that dreadful error of IQC’s that allowed me to have a look at the paper:-) In regard to the authors of the paper the only name with which I’m familiar is Raymond Laflamme who PI saddled with being interim director until Turok took over the reigns. I have the feeling he too is grateful to have more time now to dedicate to his research.

”"Entanglement" however, in contrast to Born's rule, is not an axiom of quantum mechanics, rather, it's a consequence.”

Yes and yet however is it not interesting that an axiom which itself is dictated by uncertainty mandates the certainty of correlations beyond the explanations of those possible within chance defined as being random. This was in effect Bell’s observation, that although QM mandates entanglement it fails in being able to expose as able to give a complete explanation of its mechanism. So the answer to how stands as being we don’t know, while the only answer to why is because. That is even though Einstein was wrong that the presence of entanglement within QM could not be a reasonable expectation of nature, its confirmed presence does have its explanation within the confines of the theory to remain incomplete.

I was wondering that as well. However I was more excited by the fact I could actually understand the paper compared to the other papers I read which I get about a page and am going "what the hell?". I did wonder what kind of theory would lead to those higher order corrections, but I guess they just supposed they were there and tested that.

Bee, I think you should expand the first paragraph of this into a whole blog posting. When I read that, it occurred to me that I have not read an explanation of quantum mechanics for lay-people that focuses on Born's rule as the heart of the matter.

In fact, when I read the opening phrase "If you know one thing about quantum mechanics" ... I immediately started to mentally complete this sentence, and what came to mind was:

"...the cat is both dead and alive.""...the particle goes through both slits.""...spooky action at a distance.""...there are infinite parallel universes."

Chances are that if someone really does know (or thinks they know) only one thing about quantum mechanics, it's probably something like one of the above statements (which I'm sure would cause much eye-rolling among physicists). So how about a more in-depth explanation of why Born's rule is the crux of the matter?

What should we conclude if planetary systems show clear evidence for quantized planetary spacing and dynamics?

For instance, if the Bode-Titus Law is not a "fluke", but rather is more the norm for multi-planet systems, what do we make of that?

See NASA data release + news conference on Thursday, 8/26/10.

The debate over the true nature of the structure and dynamics of Atomic Scale systems is very far from over. Bohr won the the early rounds, but I predict that Einstein will be the ultimate victor, and his insistence on rationality, causality and determinism [in the sense of dynamical systems theory] will be vindicated, albeit with some surprises added.

There must be about ten-thousand introductions to quantum mechanics on a layman level that have done that already. Sean Carroll for example in his book from eternity to here does a pretty good job, so does Feynmann in his book QED. In any case, maybe I'll come back to this and indeed write a longer post. But the point is very simple, the examples that you mention are statements about wave-functions. You need Born's rule to tell you how you get experimentally testable information, something "real," from the wavefunction. What it gives you is a probability. That's the heart of quantum mechanics because it tells you the fundamental quantity is *not* realist. We're looking back on a century of discussion about that issue. Best,

There is a website and a book that give the clearest introductions to QM I have ever read, bearing in mind I have only read a hundred or so, out of the 10,000 perhaps that exist.

They are:

Website: What is Reality? by Dr. Andrew Thomas, English electrical engineer, specifically the 1st six pages. It's a brilliant summation of Roger Penrose's "The Road to Reality" book, with clear explanations by Andrew and his excellent personal flair for making the "difficult to understand" very assesible.

The book, also from the UK, is by Tony Hey (who worked with Feynman) and Patrick Walters, The New Quantum Universe (2009 Revised and Updated Edition), which has an insanely good number of applications of QM as well. Both are beautifully illustrated with many diagrams and photographs. My only beef with Hey and Walters is that under the section for quantum gravity, they only mention String Theory, giving it 2 pages out of hundreds. Many mainstreamers would say that's the proper amount of attention QG deserves which, ouch, hurts. You'll have to order that book from Amazon as I've never seen it on an American bookstore shelf.

Peter Woit does a very nice job as well in his book "Not Even Wrong", and the very WORST is the tenure-denied cranky professor who taught us Engineers QM in undergradute school. We had tons of questions for him regarding Uncertainty, and his attitude was essentially shut up and calculate. One wonders how short the straw must be in a Physics dept. to have to teach the Engineers: Intro to Physics II. :-)

“There must be about ten-thousand introductions to quantum mechanics on a layman level that have done that already.”

I would say the best way for a laymen to come to know QM in a general way is to read two books, The first being “The Feynman Lectures on Physics: Volume Three” as it introduces the basic concepts along with the working version of the Born rule known as the path integral approach (sum over histories). The second book is David Albert’s “Quantum Mechanic and Experience” which draws closer attention to the logic aspect of the axioms of the theory to give the reader a firmer grasp of what has QM not to be a classical theory and what the implications of this turn out to be in terms of what can be said or not said with certainty from an interpretations perspective in respect to what experiment has thus far told us. I’ve always thought both of these should be prerequisites even for professional physicists in terms of providing a sound foundation from which to be able to explore things further.

Perhaps many would argue neither of these books being suitable for the laymen, yet I would point out that Feynman’s lectures were prepared for and presented to freshman, while Albert’s book doesn’t introduce complex math yet depends solely on fundamental logic. From my perspective anything less has only people able to know at best some facts about the theory with no way for them to come to be familiar with its most basic concepts to then be better able to apply critical thinking.

"Entanglement" however, in contrast to Born's rule, is not an axiom of quantum mechanics, rather, it's a consequence.

Yes for sure, following the history to today, most assuredly spooky action at a distance has to have consequences.:)

Cryptography? Quanglement?

Joel Rice,

but one would have to entertain paths that can be bent and stretched.

Yes, as much as we understand lensing, or, a gravitational bend to the photon's journey, thus being geometrical in nature, ultimately, this has a colorimetric view, of the deviations in the gravitational field?

Is it so subtle that "the thoughts" of a pretty girl and hot stove might not have some basis in such a scale, as to time fleeting, or, the other, quite the duration?

Just thinking beyond the parameters we have set for ourselves scientifically? As a layman, the thinking fallen short of those correct parameters?

From reading Feynman's QED it is clear that he is adding, multiplying and taking norms of complex numbers. (why physicists insist on saying it's the square is beyond me). The probability interpretation rests on making sure that all the possibilities are listed, with their amplitudes. Since we do not challenge the consistency of Complex Arithmetic the only question is the completeness of the possibilities. How one could know it is incomplete is a mystery to me. What does seem clear is that particles are actually oscillators with a complex phase, although Feynman does not say so explicitly, saying the complex number is 'associated' with the photon. Plato - you lost me with the colorimetry. I was just fiddling with the idea of how one would extend QED by imagining a gigantic double slit experiment where it takes billions of years for photons to arrive at the slits, while the universe is expanding.

I mean, I understand this is August, and therefore north of the Equator, deep summer, and so VACATION month (for example, I'm in Cape Cod now 300 miles from my house), so the "news" is kind of slow, but really, does that mean we reorganize our minds and "get back to basics" so to speak?

Maybe so. Nothing wrong with that actually, I just find it kind of amusing.

NOTE to Aussies and Kiwis of New Zealand. Is YOUR slow time February? Just curious.

Hi Joel Rice,

Very nice post. I'm curious if you were influenced by this post by The Scourge of Pilzen? It may well be the best thing he ever wrote. I refuse to print his name btw because of the things he says about Bee, but he does hit a homer from time to time IMO when he restricts himself to the basic stuff.

Hi Plato,

Quantum Entanglement forms the latter bookend (1935, EPR) of Quantum Mechanics that begins with Max Planck in 1900. It's always a worthy topic, but isn't the topic here more basic?

Hi Phil,

OK, OK, I'll bite! This is like the 234,832nd time you mentioned Feynman's volume II. Although I suffer from Science Book Purchasing Addiction, I guess I'll give er up and buy and read it, although Wiki seems to fill in the gaps quite nicely. :-)

Yes, Christine, I DO consider QM as "basic." Very much so. We do live in a "quantum mechanical universe", do we not? I am open-minded however and am willing to listen to non-crackpot people who disagree, but I think such people would have a very tough time explaining their case.

We also live in a General Relativistic universe, on the macro- to cosmic- scale, as well you know.

And how THOSE two seemingly incompatible theories can be united ... is pretty much what Bee and I and a host of tens of thousands are all about, no? Or ... yes?

As a Professional Astronomer, I expect and suspect you think Gen Rev is the best starting place. I don't necessarily disagree. I am however very enchanted that Bee is promoting QM. MOST followers of Smolin et. al. probably feel we should start with Gen. Rev. Should we? Nobody knows.

But whatever, each should seek to become an expert in BOTH theories before pushing the absolutely amazing field of Quantum Gravity forward, and for the moment I suspect you and Bee and a host of tens of thousands (out of a worldwide population of 7 billion) are expert in each ... so we're listening.

In the end of the day it depends on each's point of view, and more importantly: The Truth.

It doesn't matter where you start, what matters is where you end up.

And my bad for neglecting South America and parts of India and Africa, all 3 of which include both hemispheres, yes.

Steven: thanks, and I saw that post this morning and saved it on the stack, but mostly influenced by Feynman's 'QED strange theory' which seems to be quite an improvement over Vol III of lectures. A. Zee is right that this is not a popularization - at least not in the usual sense. I began to get it after 3 readings,and looking at his Thesis and Hibbs and others inbetween. The only thing that bothers me is that he might have brought up that an ideal circular electron orbit needs a whole number of those complex oscillations - Woit mentioned that on p 61 of Not Even wrong, considering that both linear and angular are just about on the same algebraic footing.Tomonaga was going on about L.H.Thomas' analysis of the precession - making just such simplified analysis, in The Story of Spin. Great book. Off to read the scourge.

Zeilinger and his group have only just begun to consider the grand implications of all their work for reality and our world. Like others in their field, they had focused on entanglement and decoherence to construct our future information technology, such as quantum computers, and not for understanding reality. But the group’s work on these kinds of applications pushed up against quantum mechanics’ foundations. To repeat a famous dictum, “All information is physical.” How we get information from our world depends on how it is encoded. Quantum mechanics encodes information, and how we obtain this through measurement is how we study and construct our world.

I asked Dr. Zeilinger about this as I was about to leave his office. “In the history of physics, we have learned that there are distinctions that we really should not make, such as between space and time… It could very well be that the distinction we make between information and reality is wrong. This is not saying that everything is just information. But it is saying that we need a new concept that encompasses or includes both.” Zeilinger smiled as he finished: “I throw this out as a challenge to our philosophy friends.”

each should seek to become an expert in BOTH theories before pushing the absolutely amazing field of Quantum Gravity forward

Yes, these are absolutely prerequisites. But you will find amazing how many people out there are ready to quickly research into quantum gravity without a thorough understanding of QM and GR. It's like someone willing to do a double-twisting gymnastics jump without knowing what is to give a simple jump first. A certain and quick door to failure.

And my bad for neglecting South America and parts of India and Africa

Eh, no problem, as I said I was not being particularly serious, just making fun, but it is true nevertheless that sometimes it is easy for us to see just one side of things.

The calorimeter design for GLAST(now Fermi) produces flashes of light that are used to determine how much energy is in each gamma-ray. A calorimeter ("calorie-meter") is a device that measures the energy (heat: calor) of a particle when it is totally absorbed. CsI(Tl) bars, arranged in a segmented manner, give both longitudinal and transverse information about the energy deposition pattern. Once a gamma ray penetrates through the anticoincidence shield, the silicon-strip tracker and lead converter planes, it then passes into the cesium-iodide calorimeters. This causes a scintillation reaction in the cesium-iodide, and the resultant light flash is photoelectrically converted to a voltage. This voltage is then digitized, recorded and relayed to earth by the spacecraft's on board computer and telemetry antenna. Cesium-iodide blocks are arranged in two perpendicular directions, to provide additional positional information about the shower.http://www-glast.stanford.edu/calorimeter.html

I can see how you wouldn't want to waste the time explaining something clarified elsewhere. I suppose I've read enough of these layman's books that I am now always looking for a little more depth, yet still don't have the time to enroll in undergraduate studies. This middle ground might be too small a target audience for there to be many books. (Feynman is one of the best sources for this though, a good recommendation!)

The popularizations understandably avoid the math but your comment left me thinking: Hey, I can handle squaring complex numbers. Why haven't I seen at least a little sample run-though of the math behind a simple quantum scenario? Please take my request not as the demands of a needy blog-reader, but as a compliment on your skill at explaining these things clearly!

(If I may) -- Imagine a bunch of laymen (a Japanese heterogeneous class, in this case), trying to understand QM by themselves, including the maths behind (in this case, the Heisenberg picture), from the very basic stuff! This book was the result, and they did a nice job! The book is filled with manga/cartoon pictures, but this does not make the book a childish or too basic one, on the contrary, it goes pretty well into details. They also wrote another book on Fourier analysis, even better one than the former. I highly recommend both books if you want to go a step further from popularization accounts.

”OK, OK, I'll bite! This is like the 234,832nd time you mentioned Feynman's volume II. Although I suffer from Science Book Purchasing Addiction, I guess I'll give er up and buy and read it, although Wiki seems to fill in the gaps quite nicely. :-)”

I think you must be counting all the times I said this in respect to other histories of which most have cancelled each other out with only a few reinforced to survive to be the actual number respective of probable times . However if you attempt in any way to discover which path was taken to establish this probability distribution the probabilities will change:-)

More seriously and with no disrespect to his book QED, Volume III is the reprint of his actual original lectures on QM that take things from the beginning, while QED is a general readers book to have explained Feynman’s own theory which extends QM with his creation of a relativistic quantum field theory of electrodynamics. If you would like the better starting point Volume III is where one should begin in my humble opinion. Now moving forward QED is good in the sense as to show how QM began to be expanded in such respect and also introduces the reader to the utility of the Feynman diagrams for visualizing situations as to ease their solution.

“ In this chapter we shall tackle immediately the basic element of the mysterious behaviour in its most strange form. We choose to examine a phenomenon which is impossible, absolutely impossible , to explain in any classical way, and which has in it the heart of quantum mechanics. In reality it contains the only mystery. We cannot make the mystery go away by “explaining” how it works. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics.”

-Richard Feynman – The Feynman Lectures on Physics; Volume III (1-1)

“Many “popular” expositions of science achieve apparent simplicity only by describing different, something considerably distorted from what they claim to be describing, we have tried to achieve maximum clarity and simplicity without compromise by distortion of the truth.”

I realize I am commenting on a long-abandoned post, but I just found it via google. I'm concerned that you would link to a video by "Dr. Quantum." Irrespective of whether this particular video is full of craziness, by pointing to it you tacitly endorse Fred Alan Wolf's credentials. His involvement in "What the Bleep?" is but a taste of the myriad ways he completely distorts science to make money off of people who don't know any better. Why not find an equally informative video that doesn't point people to a "teacher" who will just fill their heads with nonsense?